In field theory, particles are waves or excitations that propagate on the
fundamental state. In experiments or cosmological models one typically wants to
compute the out-of-equilibrium evolution of a given initial distribution of
such waves. Wave Turbulence deals with out-of-equilibrium ensembles of weakly
nonlinear waves, and is therefore well-suited to address this problem. As an
example, we consider the complex Klein-Gordon equation with a Mexican-hat
potential. This simple equation displays two kinds of excitations around the
fundamental state: massive particles and massless Goldstone bosons. The former
are waves with a nonzero frequency for vanishing wavenumber, whereas the latter
obey an acoustic dispersion relation. Using wave turbulence theory, we derive
wave kinetic equations that govern the coupled evolution of the spectra of
massive and massless waves. We first consider the thermodynamic solutions to
these equations and study the wave condensation transition, which is the
classical equivalent of Bose-Einstein condensation. We then focus on nonlocal
interactions in wavenumber space: we study the decay of an ensemble massive
particles into massless ones. Under rather general conditions, these massless
particles accumulate at low wavenumber. We study the dynamics of waves
coexisting with such a strong condensate, and we compute rigorously a nonlocal
Kolmogorov-Zakharov solution, where particles are transferred non-locally to
the condensate, while energy cascades towards large wave numbers through local
interactions. This nonlocal cascading state constitute the intermediate
asymptotics between the initial distribution of waves and the thermodynamic
state reached in the long-time limit
